A second order \(\eta \)-approximation method for constrained optimization problems involving second order invex functions. (English) Zbl 1212.90307

Summary: A new approach for obtaining second order sufficient conditions for nonlinear mathematical programming problems which makes use of the second order derivative is presented. In the so-called second order \(\eta \)-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order \(\eta \)-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second order \(\eta \)-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.


90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
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[1] T. Antczak: An {\(\eta\)}-approximation approach to nonlinear mathematical programming problems involving invex functions. Numer. Funct. Anal. Optimization 25 (2004), 423–438. · Zbl 1071.90032
[2] M. S. Bazaraa, H. D. Sherali, C. M. Shetty: Nonlinear Programming. Theory and Algorithms. John Wiley & Sons, New York, 1993.
[3] C. R. Bector, B. K. Bector: (Generalized)-bonvex functions and second order duality for a nonlinear programming problem. Congr. Numerantium 52 (1985), 37–52.
[4] C. R. Bector, M. K. Bector: On various duality theorems for second order duality in nonlinear programming. Cah. Cent. Etud. Rech. Opér. 28 (1986), 283–292. · Zbl 0622.90068
[5] C. R. Bector, S. Chandra: Generalized bonvex functions and second order duality in mathematical programming. Res. Rep. 85-2. Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, 1985.
[6] C. R. Bector, S. Chandra: (Generalized) bonvexity and higher order duality for fractional programming. Opsearch 24 (1987), 143–154. · Zbl 0638.90095
[7] A. Ben-Israel, B. Mond: What is invexity? J. Aust. Math. Soc. Ser. B 28 (1986), 1–9. · Zbl 0603.90119
[8] A. Ben-Tal: Second-order and related extremality conditions in nonlinear programming. J. Optimization Theory Appl. 31 (1980), 143–165. · Zbl 0416.90062
[9] B. D. Craven: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24 (1981), 357–366. · Zbl 0452.90066
[10] M. A. Hanson: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), 545–550. · Zbl 0463.90080
[11] O. L. Mangasarian: Nonlinear Programming. McGraw-Hill, New York, 1969.
[12] D. H. Martin: The essence of invexity. J. Optimization Theory Appl. 47 (1985), 65–76. · Zbl 0552.90077
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